Prequential randomness and probability

This paper studies Dawid’s prequential framework from the point of view of the algorithmic theory of randomness. Our first main result is that two natural notions of randomness coincide. One notion is the prequential version of the measure-theoretic definition due to Martin-Löf, and the other is the prequential version of the game-theoretic definition due to Schnorr and Levin. This is another manifestation of the close relation between the two main paradigms of randomness. The algorithmic theory of randomness can be stripped of its algorithmic aspect and still give meaningful results; the measure-theoretic paradigm then corresponds to Kolmogorov’s measure-theoretic probability and the game-theoretic paradigm corresponds to game-theoretic probability. Our second main result is that measure-theoretic probability coincides with game-theoretic probability on all analytic (in particular, Borel) sets.